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A196669 The Chebyshev primes of index 3. 5
11, 19, 29, 61, 71, 97, 101, 107, 109, 113, 127, 131, 149, 151, 173, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 257, 269, 281, 307, 311, 313, 317, 347, 349, 359, 373, 383, 389, 401, 409, 419, 421, 433, 439, 461, 479, 503, 509, 557, 563, 569, 571, 607 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The sequence consists of such odd prime numbers p that satisfy li[psi(p^3)]-li[psi(p^3-1)]<1/3, where li(x) is the logarithmic integral and psi(x) is the Chebyshev's psi function.
LINKS
M. Planat and P. Solé, Efficient prime counting and the Chebyshev primes arXiv:1109.6489 [math.NT], 2011.
MAPLE
# The function PlanatSole(n, r) is in A196667.
A196669 := n -> PlanatSole(n, 3); # Peter Luschny, Oct 23 2011
MATHEMATICA
ChebyshevPsi[n_] := Log[LCM @@ Range[n]];
Reap[Do[If[LogIntegral[ChebyshevPsi[p^3]] - LogIntegral[ChebyshevPsi[p^3 - 1]] < 1/3, Print[p]; Sow[p]], {p, Prime[Range[2, 120]]}]][[2, 1]] (* Jean-François Alcover, Jul 14 2018, updated Dec 06 2018 *)
PROG
(Magma) Mangoldt:=function(n);
if #Factorization(n) eq 1 then return Log(Factorization(n)[1][1]); else return 0; end if;
end function;
tcheb:=function(n);
x:=0;
for i in [1..n] do
x:=x+Mangoldt(i);
end for;
return(x);
end function;
jump3:=function(n);
x:=LogIntegral(tcheb(NthPrime(n)^3))-LogIntegral(tcheb(NthPrime(n)^3-1));
return x;
end function;
Set3:=[];
for i in [2..1000] do
if jump3(i)-1/3 lt 0 then Set3:=Append(Set3, NthPrime(i)); NthPrime(i); end if;
end for;
Set3;
(Sage)
def A196669(n) : return PlanatSole(n, 3)
# The function PlanatSole(n, r) is in A196667.
# Peter Luschny, Oct 23 2011
(Perl) use ntheory ":all"; forprimes { say if 3 * (LogarithmicIntegral(chebyshev_psi($_**3)) - LogarithmicIntegral(chebyshev_psi($_**3-1))) < 1 } 3, 1000; # Dana Jacobsen, Dec 29 2015
CROSSREFS
Sequence in context: A080821 A175275 A094517 * A322548 A049719 A155555
KEYWORD
nonn
AUTHOR
Michel Planat, Oct 05 2011
EXTENSIONS
Corrected and extended by Dana Jacobsen, Dec 29 2015
STATUS
approved

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Last modified May 26 13:56 EDT 2024. Contains 372826 sequences. (Running on oeis4.)